Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials

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December 2011, 31(4): 1325-1346. doi: 10.3934/dcds.2011.31.1325

Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials

Alessio Figalli ^1, and Vito Mandorino ^2,

1.	University of Texas at Austin, Department of Mathematics, 2515 Speedway Stop C1200, Austin, TX 78712-1202, United States
2.	Université Paris-Dauphine, CEREMADE UMR CNRS 7534, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France

Received November 2009 Revised August 2010 Published September 2011

Abstract
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Under a Creative Commons license

In this paper we study the properties of curves minimizing mechanical Lagrangianwhere the potential is Sobolev. Since a Sobolev functionis only defined almost everywhere, no pointwise results can be obtained in this framework,and our point of view is shifted from single curves to measures in the space of paths.This study is motived by the goal of understanding the properties ofvariational solutions to the incompressible Euler equations.

Keywords: Non-smooth Lagrangians, action-minimizing measures, Euler-Lagrange equations, value function..

Mathematics Subject Classification: Primary: 49Kxx, 49Lxx; Secondary: 70H03, 37J5.

Citation: Alessio Figalli, Vito Mandorino. Fine properties of minimizers of mechanical Lagrangians with Sobolev potentials. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1325-1346. doi: 10.3934/dcds.2011.31.1325

References:

[1]	L. Ambrosio and A. Figalli, Geodesics in the space of measure-preserving maps and plans, Arch. Ration. Mech. Anal., 194 (2009), 421-462. doi: 10.1007/s00205-008-0189-2.
[2]	L. Ambrosio and A. Figalli, On the regularity of the pressure field of Brenier's weak solutions to incompressible Euler equations, Calc. Var. Partial Differential Equations, 31 (2008), 497-509.
[3]	L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
[4]	V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits (French), Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.
[5]	Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Mat. Soc., 2 (1989), 225-255. doi: 10.1090/S0894-0347-1989-0969419-8.
[6]	P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.
[7]	C. Dellacherie and P.-A. Meyer, "Probabilities and Potential," North-Holland Mathematics Studies, 29, North-Holland Publishing Co., Amsterdam-New York, 1978.
[8]	D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.
[9]	P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.
[10]	A. I. Shnirel'man, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid (Russian), Mat. Sb. (N.S.), 128 (1985), 82-109.
[11]	A. I. Shnirel'man, Generalized fluid flows, their approximation and applications, Geom. Funct. Anal., 4 (1994), 586-620. doi: 10.1007/BF01896409.

show all references

References:

[1]	L. Ambrosio and A. Figalli, Geodesics in the space of measure-preserving maps and plans, Arch. Ration. Mech. Anal., 194 (2009), 421-462. doi: 10.1007/s00205-008-0189-2.
[2]	L. Ambrosio and A. Figalli, On the regularity of the pressure field of Brenier's weak solutions to incompressible Euler equations, Calc. Var. Partial Differential Equations, 31 (2008), 497-509.
[3]	L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000.
[4]	V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits (French), Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.
[5]	Y. Brenier, The least action principle and the related concept of generalized flows for incompressible perfect fluids, J. Amer. Mat. Soc., 2 (1989), 225-255. doi: 10.1090/S0894-0347-1989-0969419-8.
[6]	P. Cannarsa and C. Sinestrari, "Semiconcave Functions, Hamilton-Jacobi Equations, and Optimal Control," Progress in Nonlinear Differential Equations and their Applications, 58, Birkhäuser Boston, Inc., Boston, MA, 2004.
[7]	C. Dellacherie and P.-A. Meyer, "Probabilities and Potential," North-Holland Mathematics Studies, 29, North-Holland Publishing Co., Amsterdam-New York, 1978.
[8]	D. G. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Annals of Math. (2), 92 (1970), 102-163. doi: 10.2307/1970699.
[9]	P. Mattila, "Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability," Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, Cambridge, 1995.
[10]	A. I. Shnirel'man, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid (Russian), Mat. Sb. (N.S.), 128 (1985), 82-109.
[11]	A. I. Shnirel'man, Generalized fluid flows, their approximation and applications, Geom. Funct. Anal., 4 (1994), 586-620. doi: 10.1007/BF01896409.

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